Perturbation theory of dense point spectra
type: Item
instance of: scholarly article
title: Perturbation theory of dense point spectra
zbMATH Open document ID: 0646.47011
DOI: 10.1016/0022-1236(87)90038-3
author: James S. Howland
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let H be a self-adjoint operator in a separable Hilbert space with pure point spectrum. Denote by \(\{e_ n\}\) a complete orthonormal set of eigenvectors of H. The author studies perturbations of H which are small in some appropriate sense. The main result of this paper are: 1. A bounded operator A is called strongly H-finite if \(\sum^{\infty}_{n=1}| Ae_ n| <\infty\) (\(| x|\) is the norm of the vector \(x\in H)\). Let \(A_ 1,...,A_ N\) be strongly H- finite. Then for a.e. \(\kappa =(\kappa_ 1...,\kappa_ n)\in {\mathbb{R}}^ N\), the operator \(H(\kappa)=H+\sum^{N}_{j=1}\kappa_ jA^*_ jA_ j\) has pure point spectrum. It follows that \(H+\kappa <\cdot,\phi >\phi\) is pure point for a.e. \(\kappa\) if \(\sum^{\infty}_{n=1}| <\phi,e_ n>| <\infty.\) 2. Let \(X_ 1,X_ 2,..\). be a sequence of independent identically distributed (i.i.d.) absolutely continuous random variables with density \(w(x)>0\). Let \(A_ 1,A_ 2,..\). be bounded operators with \(\sum^{\infty}_{n=1}\{\sum^{\infty}_{i=1}| A_ je_ n|^ 2\}^{1/2}<\infty\). Then the random operator \(H(\omega)=H+\sum^{\infty}_{j=1}(\tanh X_ j(\omega))A^*_ jA_ j\) has pure point spectrum almost surely (a.s.). 3. Let \(\{\phi_ n\}\) be a complete orthonormal set and \(\{a_ n\}^ a \)sequence of positive numbers with lim \(a_ n=0\). If H is self-adjoint and \(\sigma(H)\) has Lebesgue measure zero, then the random compact perturbation \(H+\sum^{\infty}_{n=1}a_ n \tanh (X_ n(\omega))<\cdot,\phi_ n>\phi_ n\) is pure point a.s. 4. Let \(Y_ j(\omega)\) be i.i.d. absolutely continuous random variables with density \(g(x) > 0\). Let \(H(\omega) = \text{diag}Y_ j(\omega)\), so that the eigenvalues of the diagonal unperturbed operator \(H(\omega)\) are random variables. If \(\sum_{n=1}^\infty| Ae_ n| < \infty\) and \(W\) is a bounded operator, then \(H(\omega) + A* WA\) has pure point spectrum a.s. These results are generalized to the case where \(H\) has pure point spectrum in an open interval \(J\).
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 4054370
zbMATH Keywords: finite-rank perturbation
zbMATH Keywords: random perturbation
zbMATH Keywords: sequence of independent identically distributed absolutely continuous random variables
zbMATH Keywords: self- adjoint operator in a separable Hilbert space with pure point spectrum
zbMATH Keywords: complete orthonormal set of eigenvectors
zbMATH Keywords: perturbations
reviewed by: Ivan Vidav
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90038-3
OpenAlex ID: W2081638071
cites work: Q4606219
cites work: Unitary Equivalence Modulo the Trace Class for Self-Adjoint Operators
cites work: On the perturbation of spectra
cites work: Absence of diffusion in the Anderson tight binding model for large disorder or low energy
cites work: On equivalence of infinite product measures
cites work: Q5526485
cites work: Wave operators and similarity for some non-selfadjoint operators
cites work: Smooth operators and commutators
cites work: The abstract theory of scattering
cites work: Q3713271
cites work: Gaussian measures in Banach spaces
cites work: On a theorem of Weyl-von Neumann
cites work: Commutators and scattering theory. I: Repulsive interactions
cites work: Q4185363
cites work: Q5768777
cites work: Singular continuous spectrum under rank one perturbations and localization for random hamiltonians
cites work: On the stability of dense point spectrum for self-adjoint operators
description: scientific article; zbMATH DE number 4054370